Wolff's theory contrasts with claims that Trump's focus on Harvard is due to his son Barron's alleged rejection, which Melania Trump denies.
A White House spokesperson dismissed Wolff and The Daily Beast, accusing them of spreading fake news and asserting Trump didn't need Harvard to succeed.
Past reports, including claims from Trump's niece Mary Trump and his late sister Maryanne Trump Barry, allege Trump had someone take his college entrance exams.
Trump has recently targeted Harvard by removing federal funding and demanding information on international students, citing antisemitism as a factor.